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On the optimality of training signals for MMSE channel estimation in MIMOOFDM systems
 Junho Jo^{1} and
 Illsoo Sohn^{2}Email author
https://doi.org/10.1186/s136380150345y
© Jo and Sohn; licensee Springer. 2015
 Received: 23 December 2014
 Accepted: 26 March 2015
 Published: 16 April 2015
Abstract
In this paper, we investigate the optimality of training signals for linear minimum mean square error (LMMSE) channel estimation in multipleinput multipleoutput (MIMO) orthogonal frequency division multiplexing (OFDM) with frequencyselective fading channels. This is a very challenging problem due to its mathematical intractability and has not been analytically solved in the literature. Using the Lagrange multiplier method, we derive the optimality conditions for training signal design. Important findings revealed on optimal training signals are twofold: (i) the energies of the training signals on each subcarrier are equal, and (ii) on each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy. We verify that our results are in line with the design principles that have been derived in singlecarrier MIMO systems. Two types of optimal training signal examples that satisfy the optimality conditions are presented for practical implementations in MIMOOFDM systems. Simulation results show that the training signals based on the optimality conditions outperform other nonoptimal training signals in terms of channel estimation performance.
Keywords
 Optimal training signal
 MMSE channel estimation
 MIMO
 OFDM
 Frequencyselective fading
1 Introduction
Recently, increasing interest has been concentrated on multipleinput multipleoutput (MIMO) orthogonal frequency division multiplexing (OFDM) for broadband wireless communication. The combination of OFDM with MIMO exploits the benefits from both techniques, i.e., the robustness to combat multipath delay spread and an increase in system capacity [15]. For a practical implementation of MIMOOFDM systems, channel estimation becomes very important for the system performance. Imperfect channel estimation typically leads to the increase of error rates and reduces transmission efficiency. In this study, we focus on the problem of designing MIMOOFDM training signals for channel estimation, a critical component in many modern wireless communication systems.
Several approaches on optimal training signal design have been proposed in the literature. In the absence of prior statistical information about the channel, simple leastsquares channel estimation is used. In [6], the optimal placement of training signals is studied for a singleinput singleoutput (SISO) OFDM case. The optimal training design is extended to the MIMOOFDM case in [7]. In [8], more general training structures for MIMOOFDM are presented that utilize frequency division, time division, and code division multiplexing.
In the presence of prior statistical information about the channel, a more efficient channel estimation technique can be used. Linear minimum mean square error (LMMSE) estimators that incorporate prior knowledge to improve channel estimation are known to be optimal for this case. In [9,10], optimal conditions for training signals are studied for SISO frequencyselective fading channels. In [11], the results are extended to MIMO frequencyselective fading channels, and it is revealed that training signals across transmit antennas should be orthogonal and training signals should be equipowered. The results are very simple but effective and thus have been widely used as design guidelines for recent wireless systems. One can intuitively generalize the main principles to multicarrier systems while the optimality of the principles has remained unproved for the multicarrier systems, i.e., OFDM. For example, downlink reference signals in commercial LTE systems are designed with the same principles. In [12], an optimal training design for both leastsquares estimators and LMMSE estimators is studied assuming cyclic delay diversity OFDM systems.
Meanwhile, research interests have been shifted to more practical issues on MIMO channel estimations. In [13,14], a training signal design in the existence of inphase and quadrature imbalances is considered for SISOOFDM and MIMOOFDM systems. A robust training signal design for LMMSE channel estimator in case of imperfect knowledge of secondorder characteristics of channels is studied in [1517]. Efficient algorithms that exploit the spatial correlation of MIMO channels are proposed for training signal design in [18,19].
In this paper, we aim to complete the puzzle with the missing piece. We directly tackle a multicarrier system model and solve the optimality conditions of training signals for LMMSE estimator in MIMOOFDM systems. This is quite challenging because simultaneous considerations on all MIMO dimensions, channel statistics, multicarriers, and multisymbols lead to an extremely complex modeling and mathematically intractable problem. This has never been solved in the literature to the best of our knowledge. We analytically derive the optimality conditions using the Lagrange multiplier method, which is the principle contribution of this study.
The remainder of the paper is organized as follows. Section 2 describes the system model of our work. After Section 3 analytically derives the optimality conditions, the optimal training signal design is discussed in Section 4. Section 5 presents two types of optimal training signal examples for practical MIMOOFDM systems. Simulation results are provided in Section 6. Finally, Section 7 draws conclusions.
Notations: Uppercase and lowercase boldface letters are used for matrices and vectors, respectively. The superscript ‘ ∗’ denotes the conjugate transpose, superscript ‘T’ denotes the transpose, and superscript ‘ −1’ denotes the matrix/vector inverse. We will use \(\mathbb {E} \left [ \cdot \right ] \) for expectation, v e c(·) for matrix vectorization, t r[·] for the matrix trace, ⊗ for the Kronecker product, and I _{ N } for the N×N identity matrix.
2 System model

X _{ n }: Training signal matrix for the nth subcarrier (Q×M),

Y _{ n }: Received signal matrix for the nth subcarrier (P×M),

H _{ n }: MIMO channel matrix for the nth subcarrier (P×Q),

W _{ n }: Received noise matrix for the nth subcarrier (P×M).
where all elements of H _{ n } are uncorrelated, and all noise variables are independent, i.e., \(\mathbb {E} \left [ \mathbf {W}_{n} \mathbf {W}_{n}^{*} \right ] = \sigma ^{2} \mathbf {I}\). Aggregating the matrices for all subcarriers gives
3 Analysis on optimality conditions
In this section, the problem formulation for minimizing the LMMSE channel estimation errors in MIMOOFDM systems is presented first. Then, optimal conditions for the training signals are derived for an optimal training signal design.
3.1 Problem formulation
Our goal is to find the matrix X in the form of (5) that minimizes (10), subject to the total transmit energy constraint.
where E _{total} is the total transmit energy.
3.2 Optimality conditions
which directly follows from the fact that A A ^{−1}=I.
We first consider the diagonal parameters x in D and derive the following lemma.
Lemma 1.
where c is a constant.
Proof.
which holds for all diagonal elements in D without loss of generality. This completes the proof.
We then consider the offdiagonal parameters χ in D and derive the following lemma. Note that offdiagonal parameters are complex numbers. We split each parameter into two real parameters as χ=x+i y.
Lemma 2.
Proof.
which holds for all offdiagonal elements in D without loss of generality. This completes the proof.
4 Optimal training signal design
In this section, we provide an optimal training signal satisfying the optimality conditions derived in the previous section.
From (14) and (40), we finally reach the following theorem on the optimal training signal.
Theorem (Optimal training signal).

The energy of the training signal in each subcarrier is equal, i.e.,$$\begin{array}{*{20}l} \mathbf{tr} \left[ \mathbf{X}_{n} \mathbf{X}_{n}^{*} \right] = \frac{E_{\text{total}}}{N}, \quad \text{for} \quad n = 1,2, \ldots, N. \end{array} $$(41)

On each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy, i.e.,$$\begin{array}{*{20}l} \mathbf{X}_{n} \mathbf{X}_{n}^{*} = \frac{E_{\text{total}}}{NQ} \mathbf{I}, \quad \text{for} \quad n = 1,2, \ldots, N. \end{array} $$(42)
5 Examples of optimal training signals
In this section, we use the theorem revealed in the previous section as a design guideline and present two examples of optimal training signal implementations. These designs are practical owing to its simple structure. Note that optimal designs of training signals are not limited to the following cases.
5.1 Sequential transmission on antennas
5.2 Interlaced transmission on antennas
6 Simulation results
In this section, we verify the optimality of the training signals through extensive computer simulations. We consider a MIMOOFDM system where the number of TX antennas is Q=4, the number of RX antennas is P=4, the number of OFDM subcarriers is N=128, and the length of the cyclic prefix is 32. The number of OFDM symbols for a training signal is set to M=4. A wide sense stationary uncorrelated scattering (WSSUS) model is considered for a multipath channel [23]. A multipath intensity profile of an exponential distribution is used where the number of delay taps is L=128 and an exponentially decaying factor is α. Doppler frequency is assumed to be zero. The optimal training signal shown in Figure 1 is used in the simulations. Five nonoptimal training signals are also generated for performance comparisons. Unlike the optimal training signal that satisfies the conditions in (41) and (42), the nonoptimal training signals are created at random but they all satisfy the total transmit power E _{total}.
7 Conclusions
In this paper, optimality conditions are analytically derived and design guidelines for the optimal training signals are provided for LMMSE channel estimation for MIMOOFDM. On the basis of the analysis, we clearly reveal that the training signal that satisfies the following is optimal: (i) the energy of the training signal on each subcarrier is equal, and (ii) on each subcarrier, the training signals transmitted from the different antennas are orthogonal and of equal energy. Interestingly, the optimality conditions of training signals for LMMSE estimator in MIMOOFDM systems are basically in line with design principles known from singlecarrier MIMO systems; training signals across transmit antennas should be orthogonal and training signals should be equipowered. We mathematically prove that the simple generalization of the design principles with an additional dimension, i.e., multicarriers, still holds optimality. This work is important because the optimality conditions of training signals for LMMSE estimation in MIMOOFDM systems have been mathematically proved. Future research may include an extension of the results to more practical channel statistics, e.g., correlated channels and timevarying channels.
Declarations
Authors’ Affiliations
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